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Relation between Coefficient and Indices of x and y

The coefficie...

Question

The coefficients of the (r-1)th, rth, (r+1)th terms in the expansion of $(x + 1)^{n}$ are in the ratio of 1:3:5. Find both n and r ?

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Solution

$\frac{^{n} C_{r - 2}}{^{n} C_{r - 1}} = \frac{1}{3} \Rightarrow \frac{n! . (4 - 1)! (n - r + 1)!}{(r - 2)! (n - r + 2)! n!} = \frac{1}{3} \Rightarrow \frac{r - 1}{n - r + 2} = \frac{1}{3} \Rightarrow n - 4 r + 5 = 0$
$\frac{^{n} C_{r - 1}}{_{C_{r}}^{n}} = \frac{3}{5} = \frac{/ n! r! (n - r)!}{(r - 1)! (n - r + 1)! / n!} = \frac{3}{5} \frac{r}{n - r + 1} = \frac{3}{5} \Rightarrow 5 r = 3 n - 3 r + 3 \Rightarrow 3 n - 8 r + 3 = 0$
Solving & n-4r+5=0 3n-8r+3=0
n=7, r=3

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